# 2D Vector Scalar Product Calculator

The "2D Vector Scalar Product Calculator" comes in handy when you are trying to solve problems related to the scalar product of vectors. Instead of calculating the scalar product (also known as dot product) by hand, you can use this online calculator do the math for you by simply putting the components of two vectors into the calculator.

 Vector V1: Vector V2:
 Scalar Product:

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## Vector multiplication types

There are two types of vector multiplication:

1. the cross product (denoted by the symbol 'x')
2. the dot product which is also known as the scalar product (denoted by the symbol '.')

The main difference between these two types of multiplication is that the outcome of the product of the cross operation is a vector, while the product of the dot operation gives a single number i.e., only the magnitude.

## What does the scalar product of two vectors represent?

The scalar product of two vectors gives us an idea about the amount of one vector that goes in the direction of another vector.

Let us understand this with the help of an example:

For instance, if you pulled an object twenty metres at an inclined angle, there will be a vertical as well as a horizontal component to your force vector. If you calculate the scalar product in this case, it will give you the amount of force going in the direction that the object moved i.e., the direction of the displacement.

## What is the scalar product formula?

All our calculations will be performed in 2D space which means that every vector can be represented using two components:

a = [a1, a2]
b = [b1, b2]

The scalar product of two vectors can be defined as the product of the magnitude of the two vectors with the Cosine of the angle between them. If you want to calculate the angle between two vectors, you can use the 2D Vector Angle Calculator.

The formula for finding the scalar product of two vectors is given by:

a.b = |a| × |b| × cosθ

Where:

• a.b is the scalar product of two vectors namely 'a' and 'b'
• |a| and |b| are the magnitude of the vectors respectively
• θ is the angle between the two vectors

From the above formula, we can conclude that if the angle between the two vectors i.e., θ is 90 degrees then the scalar product of the two vectors will be zero (since, cos90 = 0 degree). Similarly, if the angle between the two vectors is 0 degree, the scalar product will give the maximum value i.e., the multiplication of the magnitudes only (since, cos0 = 1).

In certain cases, you can use the scalar product as a tool for finding the angle between the two vectors. At that time, the cosine will be equal to the ratio of the scalar product and the vectors' magnitude i.e.,

cosθ = a × b / (|a| × |b|)

You can get the value of the angle θ by finding the inverse of the value that you got on the right-hand side of the equation.

## Algebraic representation of the scalar product

Algebraically, the scalar product of two vectors in 2D can be represented as-

a.b = a1 × b1 + a2 × b2

Where

• a.b is the scalar product of two vectors namely 'a' and 'b',
• a1 and a2 are the magnitude of the vectors in "x" and "y" direction respectively
• b1 and b2 are the magnitudes of the vectors in "x" and "y" direction respectively

## How to use the 2D Vector Scalar Product Calculator?

The first step is to enter the components of the two vectors respectively in the required fields namely Vector V1 and Vector V2.

The first field is the value of the x-component of the vector and the second field will hold the y-component of the vector (in case of 2D vectors only).

The next step is to press enter and the calculator will do its work. It will represent the output as the scalar product of the two vectors.

Vectors may contain decimal and integers, but not functions, fractions or variables.

## Applications of the scalar product

• If you want to find out whether two vectors are perpendicular to each other or not, the scalar product is the easiest and simplest way to find out the same.
• The law of cosines can be proved using the scalar product.
• Numerous physical quantities are defined as a scalar product. Work is defined as the scalar product of displacement and force. Power is defined as the scalar product of velocity and force.

## Summary

In this modern era of technology, more formulas and their quick applications are used instead of wasting time on manual math calculations. This simple user interface in the form of a "2D Vector Scalar Product Calculator" can provide you with desired results, why would you depend on manually calculating math formulas and waste significant time?

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